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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity, not necessarily commutative.\n$R$ is *von Neumann regular* if for every $a \\in R$ there is $x \\in R$ with $a x a = a$.\n\n## Remarks\n\nIf $a x a = a$, then $e = a x$ is an [[idempotent|0N5NJZKKN3]] with $e R = a R$; so $R$ is von Neumann regular if and only if every principal right ideal is generated by an idempotent.\nA commutative ring is von Neumann regular if and only if it is [[reduced|SZHY975FFX]] of [[Krull dimension|MVGBS3J1P7]] $0$, and if and only if every $R$-module is [[flat|73RYYEVDKA]].\nEvery semisimple ring, for instance the ring of $n \\times n$ [[matrices|0VD9ZSWWJG]] over a [[division ring|0EA5BTA0CX]], is von Neumann regular.\n\nExample: a [[Boolean ring|0RCV1ZFP8G]] $R$, in which $a^2 = a$ for all $a$, is von Neumann regular with $x = a$: $a a a = a^2 = a$.\nExample: a [[field|04C5ZBG1WT]] is von Neumann regular, with $x = a^{-1}$ for $a \\ne 0$ and $x = 0$ for $a = 0$; so is any [[product|0QM2G72DXA]] of fields, componentwise.\n\nNon-example: $\\Z$ is not von Neumann regular.\nFor $a = 2$, the condition $2 x 2 = 2$ means $4x = 1$, which has no solution in $\\Z$.\n",
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